Practice MCQ Questions and Answer on Volume and Surface Area

1.

A copper rod of 1 cm diameter and 8 cm length is drawn into a wire of uniform diameter and 18 m length. The radius (in cm) of the wire, is :

• (A) $$\frac{1}{15}$$
• (B) $$\frac{1}{30}$$
• (C) $$\frac{2}{15}$$
• (D) $$15$$

Show Answer

 Volume of copper rod : $$\eqalign{ & = \left( {\pi \times \frac{1}{2} \times \frac{1}{2} \times 8} \right){\text{ c}}{{\text{m}}^3} \cr & = 2\pi \,{\text{ c}}{{\text{m}}^3} \cr} $$ Let the radius of the wire be r cm Then, volume of wire : $$\eqalign{ & = \left( \pi {r^2} \times 1800 \right) {\text{cm}}^3 \cr & = 1800\, \pi {r^2} {\text{ cm}}^3 \cr & \therefore 1800\pi {r^2} = 2\pi \cr & \Rightarrow {r^2} = \frac{2}{{1800}} \cr & \Rightarrow {r^2} = \frac{1}{{900}} \cr & \Rightarrow r = \sqrt {\frac{1}{{900}}} = \frac{1}{{30}} \cr} $$

2.

A hollow spherical metallic ball has an external diameter 6 cm and is $$\frac{1}{2}$$ cm thick. The volume of metal used in the ball is :

• (A) $$37\frac{2}{3}{\text{ c}}{{\text{m}}^3}$$
• (B) $$40\frac{2}{3}{\text{ c}}{{\text{m}}^3}$$
• (C) $$41\frac{2}{3}{\text{ c}}{{\text{m}}^3}$$
• (D) $$47\frac{2}{3}{\text{ c}}{{\text{m}}^3}$$

Show Answer

 External radius = 3 cm Internal radius = (3 - 0.5) cm = 2.5 cm Volume of the metal : $$\eqalign{ & = \left[ {\frac{4}{3} \times \frac{{22}}{7} \times \left\{ {{{\left( 3 \right)}^3} - {{\left( {2.5} \right)}^3}} \right\}} \right]{\text{ c}}{{\text{m}}^3} \cr & = \left( {\frac{4}{3} \times \frac{{22}}{7} \times \frac{{91}}{8}} \right){\text{ c}}{{\text{m}}^3} \cr & = \left( {\frac{{143}}{3}} \right){\text{ c}}{{\text{m}}^3} \cr & = 47\frac{2}{3}{\text{ c}}{{\text{m}}^3} \cr} $$

3.

V₁, V₂, V₃ and V₄ are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4 cm respectively. Some statements regarding these volumes are given below :
(i) V₁ + V₂ + 2V₃ V₄
(ii) V₁ + 4V₂ + V₃ V₄
(iii) 2(V₁ + V₃) + V₂ = V₄
Which of these statements area correct ?

• (A) 1 and 2
• (B) 2 and 3
• (C) 1 and 3
• (D) 1, 2 and 3

Show Answer

 Clearly, we have : V1 = x3, V2 = (2x)3 = 8x3 V3 = (3x)3 = 27x3 V4 = (4x)3 = 64x3 (i) V1 + V2 + 2V3 = x3 + 8x3 + 2 × 27x3 = 63x3 V4 (ii) V1 + 4V2 + V3 = x3 + 4 × 8x3 + 27 x3 = 60x3 V4 (iii) 2(V1 + V3) + V2 = 2 (x3 + 27x3) + 8x3 = 64x3 = V4

4.

A cistern of capacity 8000 litres measures externally 3.3 m by 2.6 m by 1.1 m and its walls are 5 cm thick. The thickness of the bottom is:

• (A) 90 cm
• (B) 1 dm
• (C) 1 m
• (D) 1.1 cm

Show Answer

 Let the thickness of the bottom be x cm Then, $${\mkern 1mu} {\left( {330 - 10} \right) \times \left( {260 - 10} \right) \times \left( {110 - x} \right)} = $$         $$8000 \times $$ $$1000$$ $$ \Rightarrow 320 \times 250 \times \left( {110 - x} \right) = 8000 \times 1000$$ $$\eqalign{ & \Rightarrow \left( {110 - x} \right) = \frac{{8000 \times 1000}}{{320 \times 250}} = 100 \cr & \Rightarrow x = 10\,{\text{cm}} = 1\,{\text{dm}} \cr} $$

5.

If the volume of a sphere is divided by its surface area, the result is 27 cm. The radius of the sphere is :

• (A) 9 cm
• (B) 36 cm
• (C) 54 cm
• (D) 81 cm

Show Answer

 $$\eqalign{ & \frac{{\frac{4}{3}\pi {r^3}}}{{4\pi {r^2}}} = 27 \cr & \Rightarrow r = 81\,cm \cr} $$

6.

A larger cube is formed from the material obtained by melting three smaller cubes of 3, 4 and 5 cm side. The ratio of the total surface areas of the smaller cubes and the larger cube is :

• (A) 2 : 1
• (B) 3 : 2
• (C) 25 : 18
• (D) 27 : 20

Show Answer

 Volume of the larger cube : $$\eqalign{ & = \left( {{3^3} + {4^3} + {5^3}} \right){\text{ c}}{{\text{m}}^3} \cr & = 216{\text{ c}}{{\text{m}}^3} \cr} $$ Let the edge of the larger cube be a cm $$\eqalign{ & \therefore {a^3} = 216 \cr & \Rightarrow a = 6 \cr} $$ Required ratio : $$\eqalign{ & = \frac{{6\left( {{3^2} + {4^2} + {5^2}} \right)}}{{6 \times {6^2}}} \cr & = \frac{{6 \times 50}}{{6 \times 36}} \cr & = \frac{{25}}{{18}}\,Or\,25:18 \cr} $$

7.

A sphere and a cube have equal surface area. The ratio of the volume of the sphere to that of the cube is :

• (A) $$\sqrt \pi :\sqrt 6 $$
• (B) $$\sqrt 2 :\sqrt \pi $$
• (C) $$\sqrt \pi :\sqrt 3 $$
• (D) $$\sqrt 6 :\sqrt \pi $$

Show Answer

 $$\eqalign{ & 4\pi {R^2} = 6{a^2} \cr & \Rightarrow \frac{{{R^2}}}{{{a^2}}} = \frac{3}{{2\pi }} \cr & \Rightarrow \frac{R}{a} = \frac{{\sqrt 3 }}{{\sqrt {2\pi } }} \cr} $$ $$\eqalign{ & \therefore \frac{{{\text{Volume of spere}}}}{{{\text{Volume of cube}}}} \cr & = \frac{{\frac{4}{3}\pi {R^3}}}{{{a^3}}} \cr & = \frac{4}{3}\pi {\left( {\frac{R}{a}} \right)^3} \cr & = \frac{4}{3}\pi \frac{{3\sqrt 3 }}{{2\pi \sqrt {2\pi } }} \cr & = \frac{{2\sqrt 3 }}{{\sqrt {2\pi } }} \cr & = \frac{{\sqrt {12} }}{{\sqrt {2\pi } }} \cr & = \frac{{\sqrt 6 }}{{\sqrt \pi }} \cr & \text{or, }\sqrt 6 :\sqrt \pi \cr} $$

8.

The volume of a sphere is $$2145\frac{{11}}{{21}}{\text{c}}{{\text{m}}^3}.$$   Its radius is equal to :

• (A) 7 cm
• (B) 8 cm
• (C) 9 cm
• (D) None of these

Show Answer

 $$\eqalign{ & \frac{4}{3} \times \frac{{22}}{7} \times {r^3} = \frac{{45056}}{{21}} \cr & \Rightarrow {r^3} = \left( {\frac{{45056}}{{21}} \times \frac{3}{4} \times \frac{7}{{22}}} \right) \cr & \Rightarrow {r^3} = 512 \cr & \Rightarrow r = \root 3 \of {512} \cr & \Rightarrow r = 8\,cm \cr} $$

9.

The radius of a cylindrical cistern is 10 metres and its height is 15 metres. Initially the cistern is empty. We start filling the cistern with water through a pipe whose diameter is 50 cm. Water is coming out of the pipe with a velocity of 5 m/sec. How many minutes will it take in filling the cistern with water ?

• (A) 20 min
• (B) 40 min
• (C) 60 min
• (D) 80 min

Show Answer

 Volume of cistern : $$\eqalign{ & = \left( {\pi \times {{10}^2} \times 15} \right){m^3} \cr & = 1500\pi {m^3} \cr} $$ Volume of water flowing through the pipe in 1 sec : $$\eqalign{ & = \left( {\pi \times 0.25 \times 0.25 \times 5} \right){m^3} \cr & = 0.3125\pi {m^3} \cr} $$ ∴ Time taken to fill the cistern : $$\eqalign{ & = \left( {\frac{{1500\pi }}{{0.3125\pi }}} \right) \cr & = \left( {\frac{{1500 \times 10000}}{{3125}}} \right) \cr & = 4800\,\sec \cr & = \left( {\frac{{4800}}{{60}}} \right)\min \cr & = 80\,\min \cr} $$

10.

The sum of the radius and the height of a cylinder is 19 m. The total surface area of the cylinder is 1672 m², what is the volume of the cylinder ?

• (A) 3080 m3
• (B) 2940 m3
• (C) 3220 m3
• (D) 2660 m3

Show Answer

 Let the radius of the cylinder be r and height be h Then, r + h = 19 Again, total surface area of cylinder = $$\left( {2\pi rh + 2\pi {r^2}} \right)$$ Now, $$\eqalign{ & 2\pi r\left( {h + r} \right) = 1672 \cr & \Rightarrow 2\pi r \times 19 = 1672 \cr & \Rightarrow 38\pi r = 1672 \cr & \therefore \pi r = \frac{{1672}}{{38}} = 44\,m \cr & \therefore r = \frac{{44 \times 7}}{{22}} = 14\,m \cr} $$ ∴ Height = 19 - 14 = 5 m Volume of cylinder : $$\eqalign{ & = \pi {r^2}h \cr & = \frac{{22}}{7} \times 14 \times 14 \times 5 \cr & = 22 \times 2 \times 14 \times 5 \cr & = 3080\,{m^3} \cr} $$